Lectures on the Cosmological Constant Problem
Antonio Padilla
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In one page
Antonio Padilla wrote these lectures for the X Mexican School on Gravitation and Mathematical Physics, and they are one of the clearest modern statements of why the vacuum’s energy is the deepest open question in physics. He begins by establishing that the fluctuations are real and that they weigh: the Lamb shift and the Casimir effect show they exist, and comparing the gravitational with the inertial mass of aluminium and platinum — whose vacuum-polarisation energies differ by a factor of about three — shows they gravitate, to about one part in a million million. Then comes the puzzle. Quantum field theory expects a vacuum energy at least as large as the TeV scale to the fourth power; cosmology measures something near the milli-electronvolt scale, sixty orders of magnitude smaller. Padilla’s point is that the real trouble is not the size of that gap but that the cancellation has to be redone by hand every time you add a loop or move the cut-off. He then works the escape routes, and argues for a global modification of gravity he calls sequestering.
Why it matters hereChapter 2 needs the strongest statement available, from inside mainstream field theory, that the vacuum energy is real, that it gravitates and that it is enormous — and this is it; chapter 13 uses the same lectures for the exact shape of the gap between what the vacuum should weigh and what the sky measures.
What it claims
01Vacuum fluctuations really exist, and the evidence is ordinary laboratory physics: the Lamb shift in hydrogen, whose vacuum-polarisation contribution splits states of the same energy quantum number by angular momentum, and the Casimir effect.Section 2.1, Eq. (2.5) and Figure 1
Settled physics02Those fluctuations also gravitate: for heavy nuclei such as aluminium and platinum the vacuum-polarisation contribution to the inertial mass differs by a factor of about three, yet the ratio of gravitational to inertial mass is the same for both to about one part in ten to the twelfth.Section 2.1, closing paragraph
Settled physics03The size of the mismatch, stated carefully: observation requires the renormalised cosmological constant to be no larger than the milli-electronvolt scale to the fourth power, while the finite part of the vacuum energy is at least the TeV scale to the fourth power — about ten to the sixtieth times bigger — so the counterterm and the vacuum energy must cancel to sixty decimal places.Section 2.2, Eqs. (2.9) to (2.11)
Published and peer-reviewed04The real problem is radiative instability rather than one big cancellation: the tuning has to be redone at every new loop order and every time the Wilsonian cut-off is moved, which is not how an effective field theory is supposed to behave — and phase transitions make it worse, the electroweak transition shifting the vacuum energy by about the 200 GeV scale to the fourth power and the QCD transition by about the 0.3 GeV scale to the fourth power.Sections 2.3 to 2.5
Published and peer-reviewed05Weinberg’s no-go theorem is worked through in full: given a translationally invariant vacuum with constant fields, extra matter fields cannot self-adjust to eat the vacuum energy without fine tuning — which means every serious proposal has to break one of the theorem’s stated assumptions, and naming which one is how a proposal should be judged.Section 3.2, Eqs. (3.5) to (3.9)
Published and peer-reviewed06Padilla’s own proposal, sequestering, modifies General Relativity globally rather than locally — indistinguishable from General Relativity in any local measurement — and cancels the protected matter sector’s vacuum energy at every order in perturbation theory, leaving a residual cosmological constant that is radiatively stable and is measured rather than predicted. Its signature is that dark energy must be transient, with the equation-of-state parameter only approximately minus one.Section 7.3 and Section 8
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Abstract
These lectures on the cosmological constant problem were prepared for the X Mexican School on Gravitation and Mathematical Physics. The problem itself is explained in detail, emphasising the importance of radiative instability and the need to repeatedly fine tune as we change our effective description. Weinberg's no go theorem is worked through in detail. I review a number of proposals including Linde's universe multiplication, Coleman's wormholes, the fat graviton, and SLED, to name a few. Large distance modifications of gravity are also discussed, with causality considerations pointing towards a global modification as being the most sensible option. The global nature of the cosmological constant problem is also emphasized, and as a result, the sequestering scenario is reviewed in some detail, demonstrating the cancellation of the Standard Model vacuum energy through a global modification of General Relativity.
Antonio Padilla, School of Physics and Astronomy, University of Nottingham. Lecture notes prepared for the X Mexican School on Gravitation and Mathematical Physics; arXiv:1502.05296 [hep-th], posted 18 February 2015.
(Abstract only — see the rights note above. On this site, Steven Weinberg’s founding review of the same problem is at /library/stm-5610822bc8, Jérôme Martin’s long-form treatment is at /library/stm-568da33759, and Raphael Bousso’s TASI lectures are at /library/stm-dbd527c8c2.)
The way in
https://arxiv.org/abs/1502.05296LICENCE CHECKED. The manuscript is arXiv:1502.05296, read on the arXiv abstract page and in full on 2026-09-08; the record carries the arXiv non-exclusive distribution licence rather than a Creative Commons licence, and no Creative Commons statement appears in the text, so this sheet carries the summary, the claims and the author’s own abstract and sends the reader to the source. The claims below were read against the complete 32-page manuscript and the locators use its own section and equation numbering. Exponents and inequalities are written out in words because the page is MDX. Padilla writes from the School of Physics and Astronomy, University of Nottingham; the notes were prepared for the X Mexican School on Gravitation and Mathematical Physics.
How to cite it
Antonio Padilla (2015) Lectures on the Cosmological Constant Problem. arXiv:1502.05296
Where it sits in the curriculum
What the vacuum isInertia and gravity from the vacuumThe unified picture